How can one find the chromatic number of the orientable surface S2 (the double-torus)? Does anyone know of an example which shows this chromatic number by giving an upper bound and a lower bound?
Thanks!
2026-03-25 06:35:46.1774420546
What is the chromatic number of S2?
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Let $S_g$ be the orientable sphere with $g$ handles. Then its chromatic number $\chi(g)$ is given by the Heawood number $$ h(S_g)=\left\lfloor \frac{7+\sqrt{1+48g}}{2}\right\rfloor $$ for $g\le 1$. So $\chi(S_2)=8$. For further details see for example here.
See also the wikipedia entry for Heawood conjecture.